Excitation of Plasmonic Wakefields in Multi-Walled Carbon Nanotubes: A Hydrodynamic Approach

Abstract

The interactions of charged particles with carbon nanotubes (CNTs) may excite plasmonic modes in the electron gas produced in the cylindrical graphene shells that constitute the carbon nanotube walls. These excitations have recently been proposed as a potential novel method of short-wavelength-high-gradient particle acceleration that may revolutionize particle acceleration techniques. In this chapter, we review a theory based on a linearized hydrodynamic model to describe the electronic excitations on the nanotube walls produced by a point-like charge moving paraxially through multi-walled CNTs. In this model, the plasmonic excitations on the nanotube surfaces are described considering the electron gas as two-dimensional plasma with additional contributions to the fluid momentum equation from specific solid-state properties of the gas. Analytical expressions of the excited longitudinal and transverse wakefields are derived. These general expressions are particularized for the case of single- and double-walled nanotubes, relating them with the resonant frequencies obtained from the dispersion relation. The dependence of the wakefields on the parameters of the model such as the particle velocity, the nanotube radii and the surface density is analyzed. Finally, a comprehensive discussion is presented, addressing the feasibility and potential limitations of employing the linearized hydrodynamic theory for modelling CNT-based particle acceleration.

1. Introduction

Particle accelerators have a strong impact in science, industry and economy. The conventional accelerators are based on the RF technology and are limited to gradients in the order of 100 MV/m due to a phenomenon called surface RF breakdown [1, 2]. Several alternative solutions are currently being investigated to surpass this limit, e.g. dielectric laser or wakefield accelerators [3, 4, 5, 6, 7] and plasma based accelerators [8, 9, 10, 11, 12]. In particular, in the 1980s and 1990s T. Tajima and others [13, 14, 15] proposed the solid-state wakefield acceleration using crystals as a potential technique to sustain TV/m acceleration gradients. In the Tajima’s original conceptual scheme [13], a longitudinal wakefield is excited by high-energy ($\approx 40$ keV) X-rays which are injected into a crystal lattice at the Bragg angle creating the Borrmann-Campbell effect [16, 17]. In the same way, ultrashort charged particle bunches are able to excite electric wakefields transforming the energy loss of the driving bunch into an increment of energy of a properly injected witness bunch. However, due to the natural crystals’ angstrom-sized channels, the beam intensity acceptance and dechanneling rate are limited.

As a hollow structure, CNTs may be used for channeling and steering charged particles similarly to crystal channeling. For instance, experimental evidences on the transport of 2 MeV ${\mathrm{He}}^{+}$ ions [18] and 300 keV electrons [19] in CNTs have been reported. Thus, CNTs can achieve wider channels in two dimensions and longer dechanneling lengths compared to natural crystals [20, 21], which make them a robust alternative for TV/m acceleration due to their exceptional electronic properties, greater flexibility, and thermo-mechanical strength. As a consequence, carbon nanostructures (CNTs or even graphene layers) are currently being studied for wakefield acceleration [22, 23, 24].

The collective oscillation of the free electron gas on the nanotubes surfaces (often referred to as plasmons) can be responsible of excitation of wakefields in CNTs. This wake effect is produced because of the interaction between the driving bunch and the CNT surface. The electronic excitations on single-or multi-walled CNTs surfaces have been theoretically studied employing a dielectric theory [25, 26, 27], a hydrodynamic model [28, 29, 30, 31], a two-fluid model [32, 33], a quantum hydrodynamic model [34], a kinetic model [35, 36, 37, 38] and a combination of a semi-classical kinetic model with a Molecular Dynamics method [39].

In this chapter we have adopted the linearized hydrodynamic model (LHM) to derive analytical expressions of the excited wakefields in MWCNTs. These expressions are particularized for the cases of SWCNTs and DWCNTs and related with the resonant frequencies. The LHM is selected because of its simplicity, its good agreement with the dielectric formalism in random-phase approximation [29] and its capability to explain the modifications inflicted on SWCNTs irradiated with swift heavy ions [40].

This chapter is organized as follows. Section 2 reviews the LHM for a MWCNT and provides general expressions for both longitudinal and transverse wakefields excited by the interaction of a charged particle that is traveling paraxially along a MWCNT. These expressions are particularized and analyzed in detail for the case of a SWCNT and a DWCNT in Sections 3 and 4, respectively. Atomic units are used throughout the manuscript, unless otherwise indicated. Eventually, a summary of the findings of this study is presented in Section 5.

2. Linearized hydrodynamic model

In this section the LHM is used to describe excitations produced by a single charged particle on the nanotube surfaces of a MWCNT [31]. A MWCNT is composed of N concentric cylinders with radii $a_1<a_2<\cdots <a_N$. In this theory, each nanotube surface is modeled as an infinitesimally thin and infinitely long cylindrical shell. The delocalized electrons of the carbon ions are considered as a two-dimensional Fermi gas (i.e. a free-electron gas) that is confined over the cylindrical walls of the MWCNT with a uniform surface density $n_0$. Unless otherwise indicated, we will assume that the surface electron density of each wall is $n_0=n_g$, where $n_g=4\times 0.107$ is the electron-gas density of a graphite sheet, which corresponds to four free electrons per carbon ion [29, 41]. We consider that a driving point-like charge Q is moving parallel to the z-axis inside the MWCNT with a constant velocity v, as illustrasted in Figure 1. Thus, if energy loss is neglected, the position of Q in cylindrical coordinates as a function of time t is given by ${\mathbf{r}}_0\left(t\right)=\left(r_0,{\phi }_0,\mathit{vt}\right)$. This driving charge Q produces a perturbation in the homogeneous electron gas on the layers, which are modeled as charged fluids with velocity fields ${\mathbf{u}}_j\left({\mathbf{r}}_j,t\right)$ and surface density $n\left({\mathbf{r}}_j,t\right)=n_0+n_j\left({\mathbf{r}}_j,t\right)$, where ${\mathbf{r}}_j=\left(a_j,\phi ,z\right)$ are the cylindrical coordinates of a general point at the surface of the jth tube and $n_j\left({\mathbf{r}}_j,t\right)$ is its perturbed density per unit area. In the LHM, fluid velocities ${\mathbf{u}}_j$ and perturbed densities $n_j$ are considered relatively small perturbations. The radial component of the velocity fields ${\mathbf{u}}_j$ is zero since the electron gas is confined to the cylindrical surfaces. To analyze wakefield dynamics, it is permissible to disregard ionic motion [42, 43], given that the time scale associated with it is orders of magnitude slower than that of electronic motion, owing to the significantly greater mass of carbon ions compared to electrons.

Figure 1.

(a) Schematic of a charge Q moving parallel to the z-axis inside a MWCNT with N = 2 cylinders. (b) Schematic of a hexagonal lattice of a DWCNT.

In the LHM, electronic excitations on the tube surfaces are described by the continuity equation

$$\frac{\partial n_j\left({\mathbf{r}}_j,t\right)}{\partial t}+n_0{\nabla }_j\cdot {\mathbf{u}}_j\left({\mathbf{r}}_j,t\right)=0,$$E1

and the non-relativistic momentum-balance equation of the electron fluid at each tube wall

$$\frac{\partial {\mathbf{u}}_j\left({\mathbf{r}}_j,t\right)}{\partial t}={\nabla }_j\mathrm{\Phi }\left({\mathbf{r}}_j,t\right)-\frac{\alpha }{n_0}{\nabla }_j{n}_j\left({\mathbf{r}}_j,t\right)+\frac{\beta }{n_0}{\nabla }_j\left[{\nabla }_j^2{n}_j\left({\mathbf{r}}_j,t\right)\right]-\gamma {\mathbf{u}}_j\left({\mathbf{r}}_j,t\right),$$E2

where we have retained only the first-order terms in $n_j$ and ${\mathbf{u}}_j$. In these equations, $\mathbf{r}=\left(r,\phi ,z\right)$ is the position vector, $\nabla =\widehat{\mathbf{r}}\frac{\partial }{\partial r}+\widehat{\mathit{\phi }}\frac{1}{r}\frac{\partial }{\partial \phi }+\widehat{\mathbf{z}}\frac{\partial }{\partial z}$, ${\nabla }_j=\widehat{\mathit{\phi }}\frac{1}{a_j}\frac{\partial }{\partial \phi }+\widehat{\mathbf{z}}\frac{\partial }{\partial z}$ differentiates only tangentially to the jth tube wall and $\mathrm{\Phi }$ is the electric scalar potential. The sum of four different contributions is shown in Eq. (2). The first term in the right-hand side is the force on electrons on the jth nanotube surface due to the tangential component of the electric field. The total electric potential is a solution of the 3D Poisson equation in free space and can be expressed as $\mathrm{\Phi }={\mathrm{\Phi }}_0+{\mathrm{\Phi }}_{\mathrm{ind}}$, where ${\mathrm{\Phi }}_0=\frac{Q}{‖\mathbf{r}-{\mathbf{r}}_{\mathbf{0}}‖}$ is the Coulomb potential generated by the driving charge and ${\mathrm{\Phi }}_{\mathrm{ind}}$ is the potential created by the perturbation of the electron fluids:

$${\mathrm{\Phi }}_{\mathrm{ind}}\left(\mathbf{r},t\right)=-\sum _j\int {\mathrm{d}}^2{\mathbf{r}}_j^{\prime }\frac{n_j\left({\mathbf{r}}_j^{\prime },t\right)}{‖\mathbf{r}-{\mathbf{r}}_j^{\prime }‖},$$E3

where ${\mathbf{r}}_j^{\prime }=\left(a_j,{\phi }^{\prime },z^{\prime }\right)$ and ${\mathrm{d}}^2{\mathbf{r}}_j^{\prime }=a_j\mathrm{d}{\phi }^{\prime }\mathrm{d}{z}^{\prime }$. The second and third terms are related to the parts of the internal interaction force in the electron gas. Concretely, the second term is introduced to take into account the possible coupling with acoustic modes defining the parameter $\alpha =v_F^2/2$ (in which $v_F={\left(2\pi n_0\right)}^{1/2}$ is the Fermi velocity of the two-dimensional gas), whereas the third term with $\beta =\frac{1}{4}$ comes from the functional derivative of the Von Weizsacker gradient quantum correction in the equilibrium kinetic energy of the electron fluid [44], describing single-electron excitations in the electron gas. The last term represents a frictional force on electrons due to scattering with the ionic-lattice charges, although may be also used as a phenomenological parameter to take into account the broadening of the plasmon resonance in the excitation spectra of different materials [25], where $\gamma$ is the friction parameter.

The Eqs. (1)–(3) can be solved by performing a Fourier-Bessel (FB) transform. In particular, the FB transform $\tilde{f}\left(m,k,\omega \right)$ of an arbitrary function $f\left(\phi ,z,t\right)$ is defined as

$$f\left(\phi ,z,t\right)=\sum _{m=-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{\mathrm{d}k}{{\left(2\pi \right)}^2}{\int }_{-\infty }^{\infty }\frac{\mathrm{d}\omega }{2\pi }{\mathrm{e}}^{\mathit{ikz}+\mathit{im\phi }-\mathit{i\omega t}}\tilde{f}\left(m,k,\omega \right).$$E4

Thus, the Coulomb potential can be expressed as

$$\frac{1}{‖\mathbf{r}-{\mathbf{r}}^{\prime }‖}=\sum _{m=-\infty }^{\infty }{\int }_{-\infty }^{\infty }\frac{\mathrm{d}k}{{\left(2\pi \right)}^2}{\mathrm{e}}^{\mathit{ik}\left(z-z^{\prime }\right)+\mathit{im}\left(\phi -{\phi }^{\prime }\right)}g\left(r,r^{\prime };m,k\right),$$E5

where $g\left(r,r^{\prime };m,k\right)\equiv 4\pi I_m\left(|k|r_{\min }\right)K_m\left(|k|r_{\max }\right)$ with $r_{\min }=min\left(r,r^{\prime }\right)$, $r_{\max }=max\left(r,r^{\prime }\right)$ and $I_m\left(x\right)$ and $K_m\left(x\right)$ are the modified Bessel functions of integer order m. If the relation (5) is introduced in (3), the FB transform of ${\mathrm{\Phi }}_{\mathrm{ind}}$ can be expressed as

$${\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\left(r,m,k,\omega \right)=-\sum _{j}g\left(r,a_j;m,k\right)a_j{\tilde{n}}_j\left(m,k,\omega \right),$$E6

where ${\tilde{n}}_j$ is the FB transform of the perturbed denisty $n_j$. If the continuity equation is utilized to eliminate ${\mathbf{u}}_j$ in (2) and the FB definition is applied, the FB transform of the perturbed densities can be obtained by solving the system of N coupled linear equations:

$$S_j\left(m,k,\omega \right){\tilde{n}}_j\left(m,k,\omega \right)-\sum _l{G}_{\mathit{jl}}\left(m,k\right){\tilde{n}}_l\left(m,k,\omega \right)=B_j\left(m,k,\omega \right),$$E7

where we have defined the following functions

$$S_j\left(m,k,\omega \right)=\omega \left(\omega +\mathit{i\gamma }\right)-\alpha \left(k^2+m^2/a_j^2\right)-\beta {\left(k^2+m^2/a_j^2\right)}^2,$$E8

$$G_{\mathit{jl}}\left(m,k\right)=n_0{a}_l\left(k^2+m^2/a_j^2\right)g\left(a_j,a_l;m,k\right),$$E9

$$B_j\left(m,k,\omega \right)=-n_0\left(k^2+m^2/a_j^2\right){\tilde{\mathrm{\Phi }}}_0\left(a_j,m,k,\omega \right).$$E10

and the FB transform of the Coulomb potential created by the driving point-like charge is given by

$${\tilde{\mathrm{\Phi }}}_0\left(r,m,k,\omega \right)=2\mathit{\pi Qg}\left(r,r_0;m,k\right)\delta \left(\omega -\mathit{kv}\right)exp\left(-\mathit{im}{\phi }_0\right),$$E11

where $\delta$ is the Dirac delta. It is interesting to note that Eq. (7) can be expressed as the following matrix equation:

$$M\overrightarrow{\tilde{n}}=\overrightarrow{B},M_{\mathit{jj}}=S_j-G_{\mathit{jj}},M_{\mathit{ij}}=-G_{\mathit{ij}}\left(i\ne j\right).$$E12

Therefore, the resonant frequencies of the collective excitations in the fluids can be calculated by solving the eigenvalue equation $det\left(M\right)=0$ for $\gamma =0$.

On the other hand, the induced longitudinal and transverse electric wakefields are, respectively,

$$\begin{array}{ll}{W}_{z,\mathrm{ind}}=-\frac{\partial {\mathrm{\Phi }}_{\mathrm{ind}}}{\partial z}& =\frac{1}{{\left(2\pi \right)}^3}\sum _{m=-\infty }^{+\infty }{e}^{\mathit{im\phi }}{\int }_{-\infty }^{+\infty }\mathrm{d}kk\left(Re\left[{\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\left(r,m,k,\mathit{kv}\right)\right]sin\left(\mathit{k\zeta }\right)\right.\\ & +Im\left[{\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\left(r,m,k,\mathit{kv}\right)\right]cos\left(\mathit{k\zeta }\right))=W_{z,Re}+W_{z,Im},\end{array}$$E13

$$\begin{array}{ll}{W}_{r,\mathrm{ind}}=-\frac{\partial {\mathrm{\Phi }}_{\mathrm{ind}}}{\partial r}& =-\frac{1}{{\left(2\pi \right)}^3}\sum _{m=-\infty }^{+\infty }{e}^{\mathit{im\phi }}{\int }_{-\infty }^{+\infty }\mathrm{d}k\left(Re\left[{\partial }_r{\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\left(r,m,k,\mathit{kv}\right)\right]cos\left(\mathit{k\zeta }\right)\right.\\ & -Im\left[{\partial }_r{\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\left(r,m,k,\mathit{kv}\right)\right]sin\left(\mathit{k\zeta }\right))=W_{r,Re}+W_{r,Im},\end{array}$$E14

where we have defined a comoving coordinate $\zeta =z-\mathit{vt}$ and used the following properties: $Re\left[\tilde{f}\left(k,\mathit{kv}\right)\right]=Re\left[\tilde{f}\left(-k,-\mathit{kv}\right)\right]$, $Im\left[\tilde{f}\left(k,\mathit{kv}\right)\right]=-Im\left[\tilde{f}\left(-k,-\mathit{kv}\right)\right]$ for the functions $\tilde{f}\left(k,\mathit{kv}\right)=\left\{{\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\left(r,m,k,\mathit{kv}\right),{\partial }_r{\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\left(r,m,k,\mathit{kv}\right)\right\}$; Re and Im denote the real and imaginary part, respectively. The previous integrals have been separated in two terms, which come from $Re\left[{\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\right]$ and $Im\left[{\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\right]$, respectively. A cutoff for large wavenumbers k has to be introduced in the numerical integration to reduce the computational time and prevent possible numerical artifacts.

3. Single-walled carbon nanotubes

In this section we are going to analyze the case of a SWCNT. As there is a single wall, we are not going to use the subscripts that refer to the different CNT walls and we will simply use $\tilde{n}\equiv {\tilde{n}}_1$ and $a\equiv a_1$. The considered system is illustrated in Figure 2.

Figure 2.

(a) Schematic of an arbitrary point-like charge Q traveling parallel to the z-axis inside a SWCNT. (b) Schematic model of the hexagonal lattice of the cylinder wall.

The FB transform of the perturbed density can be obtained from Eq. (12) as:

$$\tilde{n}\left(m,k,\omega \right)=\frac{-2\pi {\mathit{Qn}}_0\left(k^2+m^2/a^2\right)g\left(a,r_0;m,k\right)\delta \left(\omega -\mathit{kv}\right)exp\left(-\mathit{im}{\phi }_0\right)}{D_m\left(k,\omega \right)},$$E15

with $D_m\left(k,\omega \right)=\omega \left(\omega +\mathit{i\gamma }\right)-{\omega }_m^2\left(k\right)$, where

$${\omega }_m^2\left(k\right)=\alpha \left(k^2+m^2/a^2\right)+\beta {\left(k^2+m^2/a^2\right)}^2+{\mathrm{\Omega }}_p^2{a}^2\left(k^2+m^2/a^2\right)K_m\left(|k|a\right)I_m\left(|k|a\right)$$E16

are the plasmon dispersion curves and ${\mathrm{\Omega }}_p=\sqrt{4\pi n_0/a}$ is the plasma frequency. The FB transform of the electric potential is given by ${\tilde{\mathrm{\Phi }}}_{\mathrm{ind}}\left(r,m,k,\omega \right)=-g\left(r,a;m,k\right)a\tilde{n}\left(m,k,\omega \right)$.

Following the procedure described in the Appendix B of [23], the quantities $W_{z,Im}$ and $W_{r,Im}$ can be analytically calculated if $\gamma \to 0^{+}$:

$$\begin{array}{lll}{W}_{z,Im}\left(r,\phi ,\zeta \right)& =& \frac{-Q}{8{\pi }^2}\sum _{m=-\infty }^{+\infty }{e}^{\mathit{im}\left(\phi -{\phi }_0\right)}{k}_m{\mathrm{\Omega }}_p^2{a}^2\left(k_m^2+m^2/a^2\right)\\ & \times & g\left(a,r_0;m,k_m\right)g\left(a,r;m,k_m\right){\left|\frac{\partial Z_m}{\partial k}\right|}_{k=k_m}^{-1}cos\left(k_m\zeta \right),\end{array}$$E17

$$\begin{array}{lll}{W}_{r,Im}\left(r,\phi ,\zeta \right)& =& \frac{-Q}{8{\pi }^2}\sum _{m=-\infty }^{+\infty }{e}^{\mathit{im}\left(\phi -{\phi }_0\right)}{\mathrm{\Omega }}_p^2{a}^2\left(k_m^2+m^2/a^2\right)\\ & \times & g\left(a,r_0;m,k_m\right){\partial }_{r}g\left(a,r;m,k_m\right){\left|\frac{\partial Z_m}{\partial k}\right|}_{k=k_m}^{-1}sin\left(k_m\zeta \right).\end{array}$$E18

Here, the quantity $Z_m\left(k\right)=Re\left[D_m\left(k,\mathit{kv}\right)\right]={\left(\mathit{kv}\right)}^2-{\omega }_m^2\left(k\right)$ has been defined, where $k_m$ are the positive roots of $Z_m\left(k\right)$, i.e., the condition for plasma resonance $k_{m}v={\omega }_m\left(k_m\right)$. These equations are valid both inside and outside the SWCNT and agree with Eqs. (15) and (16) of [23] when $r<a$.

As the plasmon dispersion relation plays an important role in obtaining the resonant wavenumbers $k_m$, we will begin by analyzing it. Thus, Figure 3 depicts the dispersion curves ${\omega }_m\left(k\right)$ for several angular-momentum modes at different CNT radii. The resonant wavenumbers $k_m$ are the intersection of the $\mathit{kv}$ lines with the dispersion relation ${\omega }_m\left(k\right)$. Therefore, if the driving velocity increases, the resonant wavenumber $k_m$ (if it exists) decreases. The fundamental mode $m=0$ is the only one that may not satisfy the condition of plasma resonance for a sufficiently large velocity. From Eq. (16), it can easily be deduced that the plasma resonance cannot be satisfied if $v<\sqrt{\alpha }=v_F/\sqrt{2}$ and that the dispersion curves increase if the surface density $n_0$ increases. In general, if the resonant wavenumber $k_m$ exists, a second resonant wavenumber with a very large value may exist, but its contribution to the excitation of wakefields is totally negligible [23]. Since all the modes converge for large wavenumbers, if the resonant wavenumber is sufficiently large, then all the modes will have a similar resonant wavenumber (and a similar associated wavelength of the wakefield: ${\lambda }_m=2\pi /k_m$). It can be also seen that if the CNT radius increases, the condition of plasma resonance can be fulfilled for higher velocities (compare Figure 3(a) and (b)).

Figure 3.

Plasmon dispersion relations ${\omega }_m\left(k\right)$ for different modes m for (a) a = 0.36 nm and (b) a = 10 nm. The resonances $k_m$ are the intersection of the kv lines (plotted for v = 0.025c, v = 0.05c and v = 0.2c) with the dispersion curves ${\omega }_m\left(k\right)$.

Figure 4 depicts two examples of induced longitudinal wakefields in SWCNTs. As it can be deduced from Figure 3, the wavelength of the wakefield is smaller in case (a) than (b). Figure 4 also shows that the damping parameter produces an exponential decay and $W_{z,Re}\approx W_{z,Im}$, except near the position of the driving particle. The same approximation holds for the transverse wakefield, as seen in Figure 5(a). Therefore, the total induced wakefield can be approximated by $W_h\approx 2W_{h,Im}$ for $h\in \left\{z,r\right\}$. It is worth noting that $W_z$ and $W_r$ increase with the radial position inside the tube (or if the driving position $r_0<a$ increases) since $I_m\left(x\right)$ and $I_m^{\prime }\left(x\right)$ are increasing functions with the argument. Moreover, there is a phase offset of $\pi /2$ between the longitudinal and transverse wakefield. It means that will be periodic regions where a witness beam can experience simultaneously acceleration and focusing. On the other hand, Figure 5(b) shows that the perturbed density satisfies the linear approximation $n_1\ll n_0$ assumed in the LHM. In general, the linear approximation is always fulfilled when considering a single proton as the driver. However, as the perturbed densities are proportional to the driving charge, if we consider that the driving particle represents a macroparticle with thousands or millions of protons, the linear approximation may not be satisfied. Nevertheless, it is known that even in non-linear regime scenarios, the analytical equations describing the beam-driven wakefields in homogeneous plasmas in the linear regime hold fairly well [45]. Therefore, the LHM may be used as an estimation for non-linear cases.

Figure 4.

Induced longitudinal wakefield contributions on axis (r = 0) for a proton traveling on axis ($r_{\mathit{0}}=\mathit{0}$) for $\gamma ={\mathit{10}}^{-3}{\mathrm{\Omega }}_p$ and (a) $a=\mathit{0.36}$ nm, v = 0.05c and (b) a = 10 nm, v = 0.2c. The red curves show the limit $\gamma \to {\mathit{0}}^{+}$: Eq. (17). Note that the driving proton is at $\zeta =\mathit{0}$.

Figure 5.

(a) Induced transverse wakefield contributions at $r=a/2$ and (b) perturbed density for a proton traveling on axis ($r_{\mathit{0}}=\mathit{0}$) for the parameters $a=\mathit{0.36}$ nm, v = 0.05c and $\gamma ={\mathit{10}}^{-3}{\mathrm{\Omega }}_p$. The red curves show the limit $\gamma \to {\mathit{0}}^{+}$: Eq. (18).

If we are interested in the wakefield excited on axis, the fundamental mode is the only mode that is excited. In this case, when $\gamma \to 0^{+}$, $W_{z,Im}$ has a cosenoidal behavior of the kind $W_{z,Im}=W_{z,Im}^{\mathit{amp}}cos\left(k_m\zeta \right)$, so Eq. (17) can be used to perform an optimization of both surface density $n_0$ and CNT radius a in order to obtain the highest amplitude of the longitudinal wakefield $\mid W_{z,Im}^{\mathit{amp}}\mid$ for a given driving velocity v. In particular, in SI units, the optimum CNT radius $a_{\mathit{opt}}$ in units of the plasma wavelength ${\lambda }_p=2\mathit{\pi c}/{\mathrm{\Omega }}_p$ can be approximated by $a_{\mathit{opt}}/{\lambda }_p\approx 0.1185{\beta }_v$ where ${\beta }_v=v/c$ is the velocity in units of the speed of light in vacuum (see Figure 9(a) in [23]). Hence, taking into account that ${\mathrm{\Omega }}_p=\sqrt{\frac{e^2{n}_0}{{\epsilon }_0{m}_{e}a}}$ in SI units, the optimum CNT radius is given by

$$a_{\mathit{opt}}\approx \frac{{\left(0.1185v\right)}^2{\left(2\pi \right)}^2{\epsilon }_0{m}_e}{e^2{n}_0},\text{or}{a}_{\mathit{opt}}\left[\mathrm{nm}\right]\approx 156.55\frac{{\beta }_v^2}{n_0\left[{10}^{20}{\mathrm{m}}^{-2}\right]},$$E19

where e is the elementary charge, $m_e$ the rest mass of the electron, and ${\epsilon }_0$ the vacuum electric permittivity. Furthermore, the amplitude $\mid W_{z,Im}^{\mathit{amp}}\mid$ for that optimum CNT radius increases for low velocities and with the square of the surface density, as seen in Figure 6.

Figure 6.

Maximum amplitude $\mid W_{z,Im}^{\mathit{amp}}\mid$ of the longitudinal wakefield obtained for the optimum radius $a_{\mathit{opt}}$ as a function of the proton velocity v for different surface densities and $r=r_0$= 0 [23].

4. Double-walled carbon nanotubes

In this case a schematic model of the considered system was shown in Figure 1. The eigenvalue equation for a DWCNT gives the following plasmon dispersion relations [31]:

$${\omega }_{\pm }^2\left(m,k\right)=\frac{{\omega }_1^2+{\omega }_2^2}{2}\pm \sqrt{{\left(\frac{{\omega }_1^2-{\omega }_2^2}{2}\right)}^2+{\mathrm{\Delta }}^2},$$E20

where

$${\omega }_j^2=\alpha \left(k^2+m^2/a_j^2\right)+\beta {\left(k^2+m^2/a_j^2\right)}^2+n_0{a}_j\left(k^2+m^2/a_j^2\right)g\left(a_j,a_j;m,k\right)$$E21

are the dispersion relations of the individual electron fluids on the walls j = 1, 2 and

$${\mathrm{\Delta }}^2=n_0^2{a}_1{a}_2\left(k^2+m^2/a_1^2\right)\left(k^2+m^2/a_2^2\right)g^2\left(a_1,a_2;m,k\right)$$E22

arises from the electrostatic interaction between both fluids. The plasmon dispersion curves ${\omega }_{\pm }\left(m,k\right)$ for a DWCNT with $a_1=0.36$nm and $a_1=0.7$ nm are depicted in Figure 7. It can be seen that there is a splitting in the dispersion curves due to the interaction between both fluids compared to the case of a SWCNT. The frequency ${\omega }_{-}\left(0,k\right)$ exhibits a quasi-linear dispersion relation for small wavenumbers which can be expressed as ${\omega }_{-}\left(0,k\to 0\right)\approx v_{\mathrm{p}}k$ with $v_{\mathrm{p}}=\sqrt{4\pi n_0\left[a_1{a}_2/\left(a_1+a_2\right)\right]ln\left(a_2/a_1\right)+\pi n_0}$. Consequently, two different resonant wavenumbers $k_0^{\pm }$ of the fundamental mode may be excited if $v<v_p$. As the frequency ${\omega }_{+}\left(m,k\right)$ is always higher than the maximum of ${\omega }_1\left(m,k\right)$ and ${\omega }_2\left(m,k\right)$, the resonance $k_0^{+}$ may be obtained for higher velocities compared to the individual SWCNTs with radius $a_1$ and $a_2$, respectively.

Figure 7.

(a) Plasmon dispersion curves ${\omega }_{\pm }\left(m,k\right)$ for different modes m for $a_1=\mathit{0.36}$ nm and $a_2=\mathit{0.7}$ nm. For each m, the upper and lower curve correspond to the plasmon dispersion ${\omega }_{+}\left(m,k\right)$ and ${\omega }_{-}\left(m,k\right)$, respectively. (b) Fundamental modes m = 0 for $a_1=\mathit{0.36}$ nm and $a_2=\mathit{0.7}$ nm compared with the dispersion relations ${\omega }_1$ and ${\omega }_2$ of the individual electron fluids. The resonances $k_0^{\pm }$ are the intersection of the kv lines (plotted for v = 0.025c and v = 0.05c) with the dispersion curves.

In a DWCNT, the solution of Eq. (7) is given by [46]:

$${\tilde{n}}_j\left(m,k,\omega \right)=\frac{N_j\left(m,k,\omega \right)}{D_m^{+}\left(k,\omega \right)D_m^{-}\left(k,\omega \right)}\delta \left(\omega -\mathit{kv}\right),$$E23

where

$$N_1\left(m,k,\omega \right)=\left(S_2-G_{22}\right)B_1^{\prime }+G_{12}{B}_2^{\prime },N_2\left(m,k,\omega \right)=\left(S_1-G_{11}\right)B_2^{\prime }+G_{21}{B}_1^{\prime },$$E24

$$D_m^{\pm }\left(k,\omega \right)=\omega \left(\omega +\mathit{i\gamma }\right)-{\omega }_{\pm }^2\left(m,k\right),$$E25

and $B_j^{\prime }$ are the functions $B_j$, but without the Dirac delta $\delta \left(\omega -\mathit{kv}\right)$, i.e.,

$$B_j^{\prime }\left(m,k\right)\delta \left(\omega -\mathit{kv}\right)=B_j\left(m,k,\omega \right).$$E26

Similarly to the case of a SWNCT the terms $W_{z,Im}$ and $W_{r,Im}$ can be analytically integrated if $\gamma \to 0^{+}$ [46]:

$$W_{z,Im}\left(r,\phi ,\zeta \right)=\sum _{m=-\infty }^{+\infty }{e}^{\mathit{im\phi }}\left[W_{z,m}^{+}cos\left(k_m^{+}\zeta \right)+W_{z,m}^{-}cos\left(k_m^{-}\zeta \right)\right],$$E27

$$W_{r,Im}\left(r,\phi ,\zeta \right)=\sum _{m=-\infty }^{+\infty }{e}^{\mathit{im\phi }}\left[W_{r,m}^{+}sin\left(k_m^{+}\zeta \right)+W_{r,m}^{-}sin\left(k_m^{-}\zeta \right)\right],$$E28

$$W_{z,m}^{+}=\frac{k_m^{+}}{{\left(2\pi \right)}^2}{\left(Z_m^{-}\left(k_m^{+},k_m^{+}v\right){\left|\frac{\partial Z_m^{+}}{\partial k}\right|}_{k=k_m^{+}}\right)}^{-1}\sum _{j=1}^{2}g\left(r,a_j;m,k_m^{+}\right)a_j{N}_j\left(m,k_m^{+},k_m^{+}v\right),$$E29

$$W_{z,m}^{-}=\frac{k_m^{-}}{{\left(2\pi \right)}^2}{\left(Z_m^{+}\left(k_m^{-},k_m^{-}v\right){\left|\frac{\partial Z_m^{-}}{\partial k}\right|}_{k=k_m^{-}}\right)}^{-1}\sum _{j=1}^{2}g\left(r,a_j;m,k_m^{-}\right)a_j{N}_j\left(m,k_m^{-},k_m^{-}v\right),$$E30

$$W_{r,m}^{+}=\frac{1}{{\left(2\pi \right)}^2}{\left(Z_m^{-}\left(k_m^{+},k_m^{+}v\right){\left|\frac{\partial Z_m^{+}}{\partial k}\right|}_{k=k_m^{+}}\right)}^{-1}\sum _{j=1}^2{\partial }_{r}g\left(r,a_j;m,k_m^{+}\right)a_j{N}_j\left(m,k_m^{+},k_m^{+}v\right),$$E31

$$W_{r,m}^{-}=\frac{1}{{\left(2\pi \right)}^2}{\left(Z_m^{+}\left(k_m^{-},k_m^{-}v\right){\left|\frac{\partial Z_m^{-}}{\partial k}\right|}_{k=k_m^{-}}\right)}^{-1}\sum _{j=1}^2{\partial }_{r}g\left(r,a_j;m,k_m^{-}\right)a_j{N}_j\left(m,k_m^{-},k_m^{-}v\right),$$E32

where $k_m^{\pm }$ are the positive roots of $Z_m^{\pm }\left(k\right)=Re\left[D_m^{\pm }\left(k,\mathit{kv}\right)\right]={\left(\mathit{kv}\right)}^2-{\omega }_{\pm }^2\left(m,k\right)$.

Figure 8 shows the induced longitudinal wakefield generated by a proton traveling on axis in a DWCNT with $a_1=0.36$ nm and $a_2=0.7$ nm for different velocities. It can be seen that for $v=0.025c<v_p\approx 0.03c$ two resonant frequencies are excited whereas for $v=0.05c>v_p\approx 0.03c$ only the resonance $k_0^{+}$ is excited, as it can be deduced from Figure 7(b). Furthermore, it can be seen that the damping parameter produces an exponential decay and that $W_{z,Re}\approx W_{z,Im}$, except near the position of the driving particle as in the case of a SWCNT. Using the FB transform definition to ${\tilde{n}}_j\left(m,k,\omega \right)$, Eq. (23), we can obtain the perturbed densities at each wall. The perturbed densities satisfy the linear approximation ($n_j\ll n_0$), as seen in Figure 9.

Figure 8.

Induced longitudinal wakefield contributions on axis (r = 0) for a proton traveling on axis ($r_{\mathit{0}}=\mathit{0}$) at different velocity values: (a) v = 0.025c and (b) v = 0.05c. In this case, the DWCNT radii are $a_1=\mathit{0.36}$ nm and $a_2=\mathit{0.7}$ nm, and the friction parameter is $\gamma ={\mathit{10}}^{-4}{\mathrm{\Omega }}_1$ in case (a) and $\gamma ={\mathit{10}}^{-2}{\mathrm{\Omega }}_1$ in case (b), where ${\mathrm{\Omega }}_1=\sqrt{4\pi n_{\mathit{0}}/a_1}$. The red curves show the approximation for small friction: Eq. (27). Note that the driving proton is at $\zeta =\mathit{0}$.

Figure 9.

Perturbed densities normalized to $n_{\mathit{0}}$ for a proton traveling on axis ($r_{\mathit{0}}=\mathit{0}$) at different velocity values: (a) v = 0.025c and (b) v = 0.05c. The DWCNT radii are $a_1=\mathit{0.36}$ nm and $a_2=\mathit{0.7}$ nm and the friction parameter is $\gamma ={\mathit{10}}^{-4}{\mathrm{\Omega }}_1$ in case (a) and $\gamma ={\mathit{10}}^{-2}{\mathrm{\Omega }}_1$ in case (b), where ${\mathrm{\Omega }}_1=\sqrt{4\pi n_{\mathit{0}}/a_1}$.

Eqs. (29) and (30) can be used to study the amplitude that corresponds to each resonant wavenumber $k_m^{\pm }$. Thus, we are going to analyze both fundamental modes for a driving proton and $r=r_0=0$. As in this case the fundamental modes are the only excited, we denote $W_z^{\pm }\equiv W_{z,0}^{\pm }$. For instance, Figure 10(a) shows the amplitudes $W_z^{\pm }$ as a function of the proton velocity for different outer radii. It can be observed that for $a_2=0.7$ nm, $W_z^{-}$ follows a trend similar to that of a SWCNT for low velocities up to the velocity $v_p\approx 0.03c$, at which point the resonance condition cannot be further verified. Otherwise, $W_z^{+}$ produces a wider peak for $a_2=0.7$ nm. For a larger outer radius, the peak corresponding to $W_z^{-}$ shifts to higher velocities, while the peak corresponding to $W_z^{+}$ shifts to lower velocities. In the case $a_1\ll a_2$, it turns out that $k_0^{+}\approx k_0^{-}\approx k_0$ and $W_z^{+}+W_z^{-}\approx W_z^{\mathit{amp}}$, where $k_0$ and $W_z^{\mathit{amp}}$ are the resonant wavenumber and the amplitude of the wakefield, respectively, of a SWCNT with radius $a=a_1$. On the other hand, Figure 10(b) depicts the amplitude $W_z^{+}$ ($W_z^{-}$ is not excited) as a function of the inner radius for different inter-wall distances for a driving proton with velocity v = 0.9c. If the inter-wall distance is much smaller than the inner radius, the excited wakefield follows a trend similar to that of a SWCNT with a surface density of $n_0=2n_g$. Thus, DWCNTs may be used to obtain a carbon nanotube that behaves as a SWCNT with a higher surface density. Therefore, these DWCNTs may excite wakefields with greater amplitude compared to the case of SWCNTs and can be optimized using the procedure described in Section 3.

Figure 10.

(a) Amplitudes $W_z^{\pm }$ as a function of the velocity of the driving charge for different $a_2$ and $a_1=\mathit{0.36}$ nm. (b) Amplitude $W_z^{+}$ as a function of the inner radius $a_1$ for different inter-wall distances $d=a_2-a_1$ and v = 0.9c. In both cases we consider a driving proton and $r=r_{\mathit{0}}=\mathit{0}$. The results are compared with the case of a SWCNT (Eq. (17)) with (a) $a=a_1$, and (b) $a=a_1$ and $n_{\mathit{0}}=2n_g$.

5. Conclusions and outlook

The linear hydrodynamic model has been reviewed to derive general expressions for the electric wakefields excited by a point-like charge moving parallel to the axis in a MWCNT. These expressions have been particularized for the case of SWCNTs and DWCNTs, obtaining analytical expressions for small damping which have been related with the resonant wavenumbers that can be obtained from the dispersion relation.

On the one hand, the analytical expression for small damping, Eq. (17), has been utilized to optimize various parameters in SWCNTs aiming to excite the highest longitudinal wakefield, obtaining the optimum radius as given by Eq. (19). Moreover, for that optimum radius, the amplitude of the wakefield increases if the driving velocity decreases and the surface density increases.

On the other hand, in the case of DWCNTs the dispersion relation splits in two branches compared to the case of SWCNTs. Consequently, two different modes may be excited if $v<v_p$ and wakefields can be excited for a wider range of driving velocities compared to the case of SWCNTs. In particular, it has been shown that DWCNTs with small inter-wall distances compared to the inner radius behave as SWCNTs with double surface density. In this way, DWCNTs represent a potential alternative to obtain higher wakefields, which can be optimized using the expressions derived for SWCNTs.

It is important to remark that in the LHM, we are assuming the linear approximation ($n_j\ll n_0$), which has been verified to be fulfilled for a single driving proton. Furthermore, the wakefields excited by a point-like charge could be used as a Green’s function to calculate the wakefields generated by a driving bunch with an arbitrary charge distribution. However, in this case, the linear approximation may not be satisfied, although the LHM can be employed as a reasonable approximation. It is also worth mentioning that for high driving velocities we should take into account the relativistic effects which are not considered in the LHM. However, the LHM may provide a good approximation of the excited wakefields compared to particle-in-cell simulations even in some ultra-relativistic cases [23].

Finally, it has been shown that the excitation of wakefields in CNTs may be a potential alternative for particle acceleration since witness beams can simultaneously experience acceleration and focusing. It is remarkable that these wakefields can also be employed to obtain ultra-brilliant X-ray sources [47, 48]. Therefore, carbon-based nano-structures may open new possibilities for applications in particle acceleration or radiation sources.

Acknowledgments

This work has been supported by Ministerio de Universidades (Gobierno de España) under grant agreement FPU20/04958, and the Generalitat Valenciana under grant agreement CIDEGENT/2019/058.

Abbreviations

CNT	carbon nanotube
DWCNT	double-walled carbon nanotube
FB	Fourier-Bessel
LHM	linearized hydrodynamic model
MWCNT	multi-walled carbon nanotube
SWCNT	single-walled carbon nanotube